Question

Let \(b = \left[ {\begin{array}{*{20}{c}}{ - 1}\\3\\{ - 1}\\4\end{array}} \right]\), and let \(A\) be the matrix in exercise 10. Is \(b\)in the range of linear transformation\(x \mapsto Ax\)? Why or why not?

Short answer

The system represented by \(\left[ {A\,\,b} \right]\) is inconsistent; so \(b\) is not in the range of \(x \to Ax\).

Step-by-step solution

Formation of the augmented matrix

Using matrix \(A = \left[ {\begin{array}{*{20}{c}}1&3&9&2\\1&0&3&{ - 4}\\0&1&2&3\\{ - 2}&3&0&5\end{array}} \right]\) and \(b = \left[ {\begin{array}{*{20}{c}}{ - 1}\\3\\{ - 1}\\4\end{array}} \right]\), form the augmented matrix \(\left[ {A\,\,b} \right]\).

\(\left[ {\begin{array}{*{20}{c}}1&3&9&2&{ - 1}\\1&0&3&{ - 4}&3\\0&1&2&3&{ - 1}\\{ - 2}&3&0&5&4\end{array}} \right]\)

Simplification of the augmented matrix using row operations

Simplify the augmented matrix using row operations.

At row 4, multiply row 1 with 2 and add it to row 1, i.e. \({R_4} \to {R_4} + 2{R_1}\).

\(\left[ {\begin{array}{*{20}{c}}1&3&9&2&{ - 1}\\1&0&3&{ - 4}&3\\0&1&2&3&{ - 1}\\{ - 2 + 2\left( 1 \right)}&{3 + 2\left( 3 \right)}&{0 + 2\left( 9 \right)}&{5 + 2\left( 2 \right)}&{4 + 2\left( { - 1} \right)}\end{array}} \right]\)

After the row operation, the matrix will become:

\(\left[ {\begin{array}{*{20}{c}}1&3&9&2&{ - 1}\\1&0&3&{ - 4}&3\\0&1&2&3&{ - 1}\\0&9&{18}&9&2\end{array}} \right]\)

Simplification of the augmented matrix using row operations

At row 2, subtract row 2 from row 1, i.e. \({R_2} \to {R_2} - {R_1}\).

\(\left[ {\begin{array}{*{20}{c}}1&3&9&2&{ - 1}\\{1 - 1}&{0 - 3}&{3 - 9}&{ - 4 - 2}&{3 + 1}\\0&1&2&3&{ - 1}\\0&9&{18}&9&2\end{array}} \right]\)

After the row operation, the matrix will become:

\(\left[ {\begin{array}{*{20}{c}}1&3&9&2&{ - 1}\\0&{ - 3}&{ - 6}&{ - 6}&4\\0&1&2&3&{ - 1}\\0&9&{18}&9&2\end{array}} \right]\)

Interchange row 2 and row 3, i.e. \({R_2} \leftrightarrow {R_3}\).

\(\left[ {\begin{array}{*{20}{c}}1&3&9&2&{ - 1}\\0&1&2&3&{ - 1}\\0&{ - 3}&{ - 6}&{ - 6}&4\\0&9&{18}&9&2\end{array}} \right]\)

Simplification of the augmented matrix using row operations

For row 4, multiply row 3 with 3 and add it to row 4, i.e. \({R_4} \to {R_4} + 3{R_3}\).

\(\left[ {\begin{array}{*{20}{c}}1&3&9&2&{ - 1}\\0&1&2&3&{ - 1}\\0&{ - 3}&{ - 6}&{ - 6}&4\\0&{9 + 3\left( { - 3} \right)}&{18 + 3\left( { - 6} \right)}&{9 + 3\left( { - 6} \right)}&{2 + 3\left( 4 \right)}\end{array}} \right]\)

After the row operation, the matrix will become:

\(\left[ {\begin{array}{*{20}{c}}1&3&9&2&{ - 1}\\0&1&2&3&{ - 1}\\0&{ - 3}&{ - 6}&{ - 6}&4\\0&0&0&{ - 9}&{14}\end{array}} \right]\)

Simplification of the augmented matrix using row operations

At row 3, multiply row 2 with 3 and add it to row 3, i.e. \({R_3} \to {R_3} + 3{R_2}\).

\(\left[ {\begin{array}{*{20}{c}}1&3&9&2&{ - 1}\\0&1&2&3&{ - 1}\\0&{ - 3 + 3\left( 1 \right)}&{ - 6 + 3\left( 2 \right)}&{ - 6 + 3\left( 3 \right)}&{4 + 3\left( { - 1} \right)}\\0&0&0&{ - 9}&{14}\end{array}} \right]\)

After the row operation, the matrix will become:

\(\left[ {\begin{array}{*{20}{c}}1&3&9&2&{ - 1}\\0&1&2&3&{ - 1}\\0&0&0&3&1\\0&0&0&{ - 9}&{14}\end{array}} \right]\)

As the system given by \(\left[ {A\,\,b} \right]\) is inconsistent, so \(b\) is not in the range of transformation \(x \to Ax\).